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Pauli problem in dimension 4

Primarily AI-generated textHuman understanding: some partsquant-ph — Quantum Physicsmath.AT — Algebraic Topologymath.KT — K-Theory and Homology

Contributed by Dmitry Grinko ↗

SubmitterDmitry Grinko

Version 1 / Sep 29, 2026 / CC BY 4.0

Abstract

The finite-dimensional Pauli problem asks how many measurements in orthonormal bases determine every pure state of a dd-level quantum system up to a global phase. Four bases always suffice, and the answer is known to be three for d=2d=2 and four for d=3d=3 and d≥5d\geq5. Dimension four remained open because the embedding argument used in other dimensions fails there: the pure-state space CP3\mathbb{CP}^3 embeds in R9\R^9, the space of the nine independent probabilities of three bases. We show that three bases do not suffice in \(\C^4\), so exactly four are needed. The same argument shows that no ten vectors in \(\C^4\) do phase retrieval. Since eleven vectors are known to suffice, the smallest phase-retrieval frame and the smallest rank-one POVM distinguishing all pure states in \(\C^4\) both have eleven elements. All three lower bounds follow from one statement: every six-dimensional real space of traceless Hermitian 4×44\times4 matrices with a common isotropic vector, that is, a nonzero ee with e∗Te=0e^*Te=0 for all TT in the space, contains a nonzero matrix of rank at most two. We prove it with complex KK-theory: otherwise, an odd unitary map on the five-sphere would have a K1K^1-class that is nonzero by antipodal symmetry but vanishes because of the isotropic vector.

Provenance statement

This problem was "one-shotted" by ChatGPT 5.6 Sol Pro and later refined with ChatGPT 5.6 Sol Ultra, ChatGPT 6 Astra and Claude Opus 5.5.

References

  1. W. Pauli, Die allgemeinen Prinzipien der Wellenmechanik, in H. Geiger and K. Scheel (eds.), Handbuch der Physik , 2nd ed., Vol. 24, Part 1, Springer, Berlin, 1933, 83--272.
  2. A. Vogt, Position and momentum distributions do not determine the quantum mechanical state, in A. R. Marlow (ed.), Mathematical Foundations of Quantum Theory , Academic Press, New York, 1978, 365--372.
  3. L. Zambrano, L. Pereira, and A. Delgado, Minimal orthonormal bases for pure quantum state estimation, Quantum 8 (2024), Article 1244, https://doi.org/10.22331/q-2024-02-08-1244 doi:10.22331/q-2024-02-08-1244 .DOI
  4. C. Carmeli, T. Heinosaari, J. Schultz, and A. Toigo, How many orthonormal bases are needed to distinguish all pure quantum states?, Eur. Phys. J. D 69 (2015), Article 179, https://doi.org/10.1140/epjd/e2015-60230-5 doi:10.1140/epjd/e2015-60230-5 .DOI
  5. P. Jaming, Uniqueness results in an extension of Pauli's phase retrieval problem, Appl. Comput. Harmon. Anal. 37 (2014), 413--441, https://doi.org/10.1016/j.acha.2014.01.003 doi:10.1016/j.acha.2014.01.003 .DOI
  6. B. Z. Moroz and A. M. Perelomov, On a problem posed by Pauli, Theoret. Math. Phys. 101 (1994), 1200--1204, https://doi.org/10.1007/BF01079256 doi:10.1007/BF01079256 .DOI
  7. T. Heinosaari, L. Mazzarella, and M. M. Wolf, Quantum tomography under prior information, Comm. Math. Phys. 318 (2013), 355--374, https://doi.org/10.1007/s00220-013-1671-8 doi:10.1007/s00220-013-1671-8 ; corrected version https://arxiv.org/abs/1109.5478v3 arXiv:1109.5478v3 , 2018.arXivDOI
  8. D. Mondragon and V. Voroninski, Determination of all pure quantum states from a minimal number of observables, https://arxiv.org/abs/1306.1214 arXiv:1306.1214v2 [math-ph] , 2015.arXiv
  9. A. Mukherjee, Embedding complex projective spaces in Euclidean space, Bull. London Math. Soc. 13 (1981), 323--324, https://doi.org/10.1112/blms/13.4.323 doi:10.1112/blms/13.4.323 .DOI
  10. J. Finkelstein, Pure-state informationally complete and ``really'' complete measurements, Phys. Rev. A 70 (2004), 052107, https://doi.org/10.1103/PhysRevA.70.052107 doi:10.1103/PhysRevA.70.052107 .DOI
  11. A. S. Bandeira, J. Cahill, D. G. Mixon, and A. A. Nelson, Saving phase: Injectivity and stability for phase retrieval, Appl. Comput. Harmon. Anal. 37 (2014), 106--125, https://doi.org/10.1016/j.acha.2013.10.002 doi:10.1016/j.acha.2013.10.002 .DOI
  12. R. Balan, P. Casazza, and D. Edidin, On signal reconstruction without phase, Appl. Comput. Harmon. Anal. 20 (2006), 345--356, https://doi.org/10.1016/j.acha.2005.07.001 doi:10.1016/j.acha.2005.07.001 .DOI
  13. A. Conca, D. Edidin, M. Hering, and C. Vinzant, An algebraic characterization of injectivity in phase retrieval, Appl. Comput. Harmon. Anal. 38 (2015), 346--356, https://doi.org/10.1016/j.acha.2014.06.005 doi:10.1016/j.acha.2014.06.005 .DOI
  14. C. Vinzant, A small frame and a certificate of its injectivity, in 2015 International Conference on Sampling Theory and Applications (SampTA) , 197--200, https://doi.org/10.1109/SAMPTA.2015.7148879 doi:10.1109/SAMPTA.2015.7148879 .DOI
  15. Y. Wang and Y. Shang, Pure state `really' informationally complete with rank-1 POVM, Quantum Inf. Process. 17 (2018), Article 51, https://doi.org/10.1007/s11128-018-1812-2 doi:10.1007/s11128-018-1812-2 ; https://arxiv.org/abs/1711.07585 arXiv:1711.07585 .arXivDOI
  16. M. F. Atiyah, K-Theory , W. A. Benjamin, New York, 1967, https://www.maths.ed.ac.uk/ v1ranick/papers/atiyahk.pdf available online .link
  17. A. Hatcher, Vector Bundles and \(K\)-Theory , version 2.2, 2017, https://pi.math.cornell.edu/ hatcher/VBKT/VB.pdf available online .link
  18. M. W. Hirsch, Differential Topology , Graduate Texts in Mathematics 33, Springer, New York, 1976.
  19. A. Causin, On the dimension of some real spaces of bounded rank matrices, Linear Algebra Appl. 434 (2011), 501--506, https://doi.org/10.1016/j.laa.2010.09.004 doi:10.1016/j.laa.2010.09.004 .DOI
  20. M. Huang and J. Kim, Phase retrieval in \( ^4\) requires exactly eleven measurements, https://arxiv.org/abs/2607.27719 arXiv:2607.27719 , 2026; v1 by M. Huang, 30 July 2026; coauthored with J. Kim from v2, 28 August 2026; v3, 31 August 2026.arXiv

Version history

  1. v1Initial depositCurrentSep 29, 2026